Calculating Compression Time for Mass-Spring Impact: A Comparison of Methods

[VinceA]

This is a simple(?) problem but my ability to perform basic calculus disappeared long ago. I am working thru the equations for a spring response to a dropped weight. My primary reference is Shigley (for just about everything!) and I've validated the eqn's for deflection and max force. However, I am trying to work out the duration of the event for purposes of estimating simulation time in explicit FEA. Solving for time to max deflection from eqns in Shigley yields exactly 1/200th of the time reported by 2 mechanism simulation tools...who nail the deflection and force as noted above.

Has anyone solved this "compression time" and, if so, can you share the eqn?

 

[NateTG]

Not that you're giving a whole lot of info, but based on what you have listed, I would guess that:
1. You've got one side dealing with the whole colision, and the other dealing with half of it (introducing a factor of 2)
and
2. There's some sort of unit issue providing a factor of 100.

Note that this is a relatively wild guess.

 

[VinceA]

I've beat up the units issue so I don't think that's the problem. The system is pretty simple and I've been staring at it for a couple of hours so it didn't occur to me to give more detail... Essentially, I'm looking at a spring to ground with K=238.1 #/in. A weight (W) of 7.43# is being dropped a height (h) of 12 inches onto the spring.

Compression (D) = (W/k)+(W/k)*SQRT[1+((2*h*k)/W)] = 0.897 in.

Force (F) = K*D = 213.6 lbf

Both these calcs agree with the simulation results.

In Shigley "Mechanical Engineering Design", D is determined by assuming a Cosine term in a longer eqn is unity...

Cos[(kg/W)t - Phi] = 1

thus...

(kg/W)t-Phi = 0 or (kg/W)t = Phi or t = (Phi*W)/KG

Phi = Pi/2 + SQRT[atan(W/(2kh))] = 1.84

thus... t = 0.00015 sec

which is 1/100th (not 1/200th...sorry) the duration reported by simulation.

Can anyone see where I messed up? I, of course, waited until the last minute before deciding I wanted to add this to a slide in a webcast tomorrow.

Thanks!

 

[NateTG]

...is your calculator set to degrees?